Algebraic cycles and approximation theorems in real algebraic geometry

Abstract:

Let $M$ be a compact ${C^\infty }$ manifold. A theorem of Nash-Tognoli asserts that $M$ has an algebraic model, that is, $M$ is diffeomorphic to a nonsingular real algebraic set $X$. Let $H_{{\text {alg}}}^k(X,\mathbb {Z}/2)$ denote the subgroup of ${H^k}(X,\mathbb {Z}/2)$ of the cohomology classes determined by algebraic cycles of codimension $k$ on $X$. Assuming that $M$ is connected, orientable and $\dim M \geq 5$, we prove in this paper that a subgroup $G$ of ${H^2}(M,\mathbb {Z}/2)$ is isomorphic to $H_{{\text {alg}}}^2(X,\mathbb {Z}/2)$ for some algebraic model $X$ of $M$ if and only if ${w_2}(TM)$ is in $G$ and each element of $G$ is of the form ${w_2}(\xi )$ for some real vector bundle $\xi$ over $M$, where ${w_2}$ stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups $G$ of ${H^1}(M,\mathbb {Z}/2)$.